Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation
The method for defining basic geometric parameters of precision quadrupole lenses is offered. The method allows one to obtain the field of working area differing from required one less than by 10⁻⁴ compared with the basic component. The lens shape is described by 2 parameters. The method allows taki...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation / A.O. Mytsykov // Вопросы атомной науки и техники. — 2000. — № 2. — С. 76-79. — Бібліогр.: 4 назв. — англ. |
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| author | Mytsykov, A.O. |
| author_facet | Mytsykov, A.O. |
| citation_txt | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation / A.O. Mytsykov // Вопросы атомной науки и техники. — 2000. — № 2. — С. 76-79. — Бібліогр.: 4 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The method for defining basic geometric parameters of precision quadrupole lenses is offered. The method allows one to obtain the field of working area differing from required one less than by 10⁻⁴ compared with the basic component. The lens shape is described by 2 parameters. The method allows taking into account the field distortions caused by saturation of iron. Moreover, it is possible to calculate lenses with a complicated multipolar structure including the forbidden field harmonics. The paper contains the results of calculations for lense shapes used in actual installations.
|
| first_indexed | 2025-12-07T19:02:20Z |
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| fulltext |
CALCULATION OF GEOMETRIC PARAMETERS OF A MAGNETIC
QUADRUPOLE LENS WORKING NEAR TO SATURATION
A.O. Mytsykov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The method for defining basic geometric parameters of precision quadrupole lenses is offered. The method
allows one to obtain the field of working area differing from required one less than by 10
-4
compared with the basic
component. The lens shape is described by 2 parameters. The method allows taking into account the field distortions
caused by saturation of iron. Moreover, it is possible to calculate lenses with a complicated multipolar structure
including the forbidden field harmonics. The paper contains the results of calculations for lense shapes used in actual
installations.
PACS: 02.Dk, 02.30.Em.
INTRODUCTION
The multipolar magnetic lenses are ones from the
basic elements in the accelerator engineering. For
calculation of lenses the problem is set as follows:
according to the specific field distribution it is necessary
to obtain geometric parameters of a multipole. Major of
these parameters is the pole shape. The basic purpose of
this work is deriving of analytical expressions, which
connect by a relationship the pole shape with the field
multipolar structure.
APPLICATION OF CONFORMAL MAP FOR
DESCRIPTION OF MULTIPOLES WITH
CURVILINEAR POLES
As is known the magnetic field (hereinafter - field)
can be expressed in terms of the complex potential z(ω)
and in terms of the inverse to complex potential function
ω (z) by expression [1]
*1*
=
=+=
−
dz
di
d
dziiBBB yx
ω
ω
(1)
The function ω(z) can be obtained considering a
conformal mapping of a rectilinear band 0<Im(z)<H on
a “pole band” which is generated by the symmetry lines
of a multipole and by the pole profile Fig.1; [2].
( ) ( )[ ]∫=
z
z
dzzGz
0
expω (2)
where the function G(z) coincides within the accuracy of
an integration constants with the Schwarz integral for
the band [2]:
( ) ( ) ( )
( ) ( )
−−
−
−−=
∫
∫
∞
∞−
∞
∞−
dt
H
t
H
zttv
dt
H
t
H
zttv
H
zG
H
ππ
ππ
th
2
th
th
2
cth
2
1
0
(3)
where: H is the height of the rectilinear band on the
plane z; ν0(t)is the declination angle of the lower shore
of the “pole band” (symmetry lines of multipole) as a
function of the abscissa t of the z-plane (z=t+is) of the
rectilinear band; νH(t)is the declination angle of the
upper shore of the “pole band” (pole shape) as a
function of the abscissa t of the z-plane (z=t+is) of the
rectilinear band.
At this map (2) the lower shore of the band
0<Im(z)<H of the z-plane is transformed to the
symmetry line of a multipole and the upper one to the
pole profile. Then the first summand of the integral (3)
answers for the problem symmetry. Hereinafter we shall
consider only 'pure' multipoles with the rectilinear
symmetry lines. Then the following definition is right:
( )
∞∈
− ∞∈
=
],0[0
]0,[
0 t
tU
tv mult (4)
z=t+isH
a1 a2 a3 a3
Lower bank
Upper bank
α
β
A4
A3
A2
A1
ω=x+iyz(ω)
ω(z)
Umult
Fig. 1. The scheme of conformal mappings for a
multipole.
Obviously that for thedipole symmetry Umult=0; for
the quadrupole symmetry Umult =-π/2; for the sextupolar
symmetry Umult = -π2/3; etc.
With such a definition the first summand of the
integral (3) takes the form:
( ) ( )
( ).
2
exp1
ln
2
1th
2
cth0
zYmultUH
z
multU
H
dt
H
t
H
zttv
⋅=
−
−
=
∞
∞−
−−
∫
π
π
ππ
(5)
So, for the multipolar symmetry this factor
determines the growth of field from the zero value at the
aperture center to the certain value at the aperture edge.
The second part of the integral (3) describes the pole
shape. Let us assume that the pole has a rectilinear
exterior slants and an interior sector. Then the
declination angle of the tangent for the upper shore to
the abscissa axis of ω-plane varies under the law:
76 ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, № 2.
Серия: Ядерно-физические исследования (36), c. 76-79.
( )
∞∈
=
− ∞∈
=
∞
=
∞−
∑
],[
][
],[
2
0
1
atU
tqtQ
atU
tv
M
j
j
jH (6)
After simple transformations the second part of (3)
becomes as
( ) =−=
−−∫
∞
∞−
)(th
2
th)(
2
1 zTdt
H
t
H
zttv
H H
ππ
(7)
( ) +
−∞− H
zHzaSU 0,,,1 (7a)
( ) ( )( ) −
=
−∑
M
j
jHzaSjHzaSjq
0
,,,1,,,2 (7b)
( )0,,,2 HzaSU∞ , (7c)
where: the first summand (7a) is the contribution of the
left slant from а1; the second summand (7b) is the
contribution of the part between points а1 and а2; the
third summand (7c) is the contribution of the right slant
from а2.
( )
( ) .]exp[1Li
12
1]
2
exp[1Li
0 )!(
!
1
,,,
−
−++
−−−−+
×
= −+
−= ∑
H
t
kkH
zt
k
j
k kj
j
k
kHkjtjHztS
ππ
π (8)
Here and further the polylogarithmic function Lij
possessing the following properties is used:
∫ −=−+
−=−−
+−=−
.][Li][1Li
];,[Li][1Li
];1ln[][1Li
dxxej
xej
xejj
xejdx
d
xexe
(9)
DEPENDENCE BETWEEN FIELD
HARMONIC STRUCTURE AND POLE
SHAPE
The integral (2) representing the function inverse to
the complex potential is convenient to be present as:
( ) ( ) ( )[ ]∫ +=
z
z
mult dzzTzYUz
0
expω , (10)
where: Y(z), T(z) is defined by (5), (9).
Obviously that the field in coordinates of the band
0<Im(z)<H looks like (see (1)):
( ) ( ) ( )[ ]zTzYUzB mult −−= exp . (11)
This expression has as an argument a coordinate of
the z-plane instead of physical Cartesian coordinates of
the ω-plane containing a pole. Therefore expressions
(10,11) are the parametric representation of a field. In
practice it is convenient to have expressions describing a
field in physical coordinates (i.e. in coordinates of the
ω-plane). For deriving such expressions we use the
following procedure.
We shall obtain derivatives of the field dBi(z)/dω(z)i
in the ω-plane as functions of z. For this purpose we
note that the increment of coordinates on the plane ω is
equal to
( ) ( )[ ]dzzTzYUd mult += expω . (12)
Now we can obtain the first derivative of a field in
coordinates of a z-plane:
( ) ( )( )
( ) ( )( )[ ]zTzYU
zTzYU
dz
d
d
dB
mult
mult
+
−−
=
2expω
. (13)
Similarly it is possible to obtain any derivative, using
the recursive ratios:
( ) ( ) ( )[ ]
( )
( )
( )
( ) ( )( )[ ] .
exp
1
1
;exp
zTzYmultU
idw
zBid
dz
d
izd
zBid
zTzYmultUzB
+
−
−
=
−−=
ω
(14)
Let us relate the point z(0,0) of the z-plane
containing the band 0<Im(z)<H and the point ω(0,0) of
the plane containing the pole. Then the factors
calculated according to the formulas (14) are the Taylor-
series coefficients of the field but now for the ω-plane
containing the pole i.e. in physical, Cartesian
coordinates. It is very important for the practical
applications.
IDEAL QUADRUPOLE SYMMETRY
As an ideal quadrupole we shall mean a case for
which Umult =−π/2 (see (4), Fig.1). Let us consider a map
of the rectilinear band 0<Im(z)<H to the “polar band”
not laying down conditions to the pole shape. Suppose
that there is a certain “shape function” T(z) equal to the
integral on the upper shore of the band (see (7)).
The principal performance of a quadrupole lens is
the gradient
Having produced all substitutions for the known
values Umult, Y (z) we shall obtain:
( )
( )
( )
( )
.
2exp22
']exp[12
][
+
+−+−
−==
H
zzTH
zHT
H
z
zN
zd
zBd
π
ππ
ω
(15)
The solution of Eq. (26) is:
( )
+
−
−
=
∫
]
][22
]exp[1
ln[
2
1][0
dzzNC
H
z
zT
π
. (16)
Here the label T0(z) is applied to distinguish the
actual “shape function” (7) (in which both angularities
and curvature of the medial sector of the pole profile are
taken into account) from the ideal one (16). From the
conventional definition, that the field at centre of a
multipole is equal to zero follows С=0 (constant of the
integration in (16). In this case the map of the rectilinear
band of the z-plane to the polar band of the ω-plane,
looks like (substitution (16) in (10)):
( ) ∫
∫
=
z
z
dz
dzzN
z
0 ][2
1ω ; (17)
where the following definition is right:
77
+== ∑∑
==
M
i
i
i
M
i
i
i zNzNzN
1
0
0
1][ λ . (18)
In particular, if N(z) =const=N0 the expression (17)
becomes simpler to the well-known expression, the
inverse of which gives a complex potential for the
quadrupole symmetry.
( ) 20
000 2
;
0
2
2
1
0
ωω Nz
zN
zdz
zN
z
z
z
=
=
== ∫ . (19)
The coefficient N0 has a simple metric sense. It is
possible to show that by virtue of properties of
conformal representations:
.
2
;1
0
RH
H
N == (20)
In case of calculation of the lens with a complicated
multipolar composition it is necessary to solve the
following set of equations:
( )
( ) ( )
( )
( ) ( )
( )
=
=
=
+
+
=
=
=
+
+
=
=
=
+
∑
∑
∫
∫ ∑
.,
;
0
2,
!2
0Im
;0
0
2,
!2
0Re
;
0
1
102
Im
RyRxR
H
i
i
yRxR
i
iB
i
i
yRxR
i
iB
H
iH
dziH
k
iN
dz
ω
ω
λ
(21)
The first equation of this system expresses the fact
that with a conformal map the value of scalar potential
(imaginary part complex) on equipotential lines which
are mapped each other is intact. The second equation
expresses the circumstance that a certain point (Rx,Ry) of
the ω-plane is mapped to the point (0,H) of z-plane. The
last equation defines the position of the point (Rx,Ry) on
the ω plane by setting the aperture radius. The system
of four equations (21) contains 4 unknowns Rx, Ry, N0, H
and can by solved if coefficients λi are known quantities.
The coefficients λi can be obtained from consideration
of field derivatives on the rules (14) with allowance for
(18):
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
.83628802
432256325
62497664004
2
256985604
2734729
0693017
6108004233603
21127
0693013
;41202
2165
02109
;2
3
0605
;001
+
+++=
++=
+=
=
=
λλ
λλλλλ
λλλλ
λλ
λ
NB
NB
NB
NB
NB
(22)
The process can be prolonged as much as long.
However in practice the low numbers of harmonics are
most significant. The system (22) can be solved
concerning the even expansion coefficients λi. Here is
the solution for the first terms of expansion:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
.0,,;
;
08172964800
011440096096
08172964800
0560560007680
;
7484400
00154010780
;
7560
03028
;
6
0
531
12
0
292
0
925
0
12
0
451352
0
173
0
8
9
0
95
0
35132
0
6
6
0
9
0
25
4
3
0
5
2
=
−
+
−−
=
−+
=
+−
=
=
λλλ
λ
λ
λ
λ
N
BNBBN
N
BBBNBN
N
BBNBBN
N
BNB
N
B
(23)
Thus set of equations (21), (23) completely define
the ideal shape function according to the given
multipolar structure (coefficients Bi).
ACTUAL PROFILES OF QUADRUPOLE
LENSES
Now, when the forms of ideal shape functions (16)
and actual one (7) are known, we shall require their
coincidence in M points on a segment [0,H] which
corresponds to the radius of the aperture on the ω-plane.
(see (19,20)). The better is the coincidence of these
functions in a working area, the less is the deviation of
the field in actual geometry from the required field.
Therefore requirement of equality of ideal and actual
functions of the shape in M points of the working area
generates a system M of equations, which can be used
for determination of conformal map parameters.
The reasons of symmetry reduce in the requirements:
.5
5
3
314
][
;;
2
;21
++++−=
=∞−−=∞−=−=
tqtqtqtQ
UUaaa
π
ααπ
(24)
Let us require that in points of conjugation of
various profile sectors the function Q(t) was continuous.
Add to this the requirements (24) and we shall obtain a
set of equations for definition of another parameters.
==−=
−==
===
−−===−
+
∞
=
∞−
=
∑
∑
3,2,1,0,
4
;
;][
;
2
][
120
12
0
0
iqq
aaa
UtqaQ
UtqaQ
i
M
i
i
i
M
i
i
i
π
α
απ
(25)
Thus if, for example, we tried to specify Q(t) by the
polynomial of 7-th degree, then only 2 parameters will
be by free. Hereinafter we shall use the qi parameters at
the lowest degrees of a polynomial as free ones. The
coefficients at the higher degrees will answer for
conjugation of the pole surface.
78
Further the lens with the aperture 35 mm and with
the slant angle 30° was considered, Fig. 2. This lens is
planed to be used in the installation [4]. The parameters
a, q1, q3 were defined. The pole profile constructed on
these parameters (2) was substituted in the program
MERMEID [3] as an input. The multipolar composition
of the field was determined at various currents, Fig. 3.
The harmonics presented in the expansion are caused
by the influence of the excitation coil and saturation
effects. Therefore calculated multipolar composition
was substituted in (23) with the opposite sign for
definition of preliminary changed parameters of
conformal transformation. As a result a new profile of
the pole was obtained. The allocation of the gradient for
the lens with such a pole is shown in Fig. 4.
Fig. 2. Cross-section of quadrupole lens.
Fig. 3. The radial dependence of the gradient in the
lens (R=0.035 m) with the pole shape obtained by the
model based on the conformal map for various gradient
values.
CONCLUSION
The analytical expressions, obtained in this paper, at
shared use with programs of numerical simulation
allow to obtain by 1-2 iterations the design solution for a
quadrupole magnet, having eliminated from operation a
phase "of creative searching" of designer. The
calculation results are directly applicable for synthesis
of profiles of quadrupole lenses "burdened" by even
harmonics. The possibilities of a method do not
eliminate operation with forbidden, odd harmonics too.
For this purpose it is necessary to take into account the
curvature of lines corresponding to lines of symmetry of
an ideal quadrupole (i.e. lower coast of the band) and to
conduct evaluations for each quadrant.
Fig. 4. The radial dependence of the gradient in the
lens (R=0.035 m) with the preliminary changed pole for
various gradient values.
REFERENCES
1. М..А. Lavrent’ev, B .V .Shabat Methods of the
theory of complex variable function, 1973, M.
Nauka, 736 p. (in Russian).
2. А. Mytsykov. Application of conformal represen-
tation for model operation of flat fields created by
smooth poles. Problems of Atomic Scince and
Technology, 1999. Ser.-NPE (33), No 1, p. 102-103
(in Russian).
3. E. Bulyak, A. Dovbnya, P. Gladkikh et al. A
multipurposal accelerator facility for the Kharkov
National Scientific Center. Proceedings of
International Synchrotron Radiation Conference
Novosibirsk-98.
4. Mermaid Users's Guide. Sim Limited, Novosibirsk,
1994.
79
Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation
A.O. Mytsykov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
Introduction
Application of conformal map for description of multipoles with curvilinear poles
Dependence between field harmonic structure and pole shape
Ideal quadrupole symmetry
Actual profiles of quadrupole lenses
CONCLUSION
References
|
| id | nasplib_isofts_kiev_ua-123456789-82287 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T19:02:20Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Mytsykov, A.O. 2015-05-27T14:01:46Z 2015-05-27T14:01:46Z 2000 Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation / A.O. Mytsykov // Вопросы атомной науки и техники. — 2000. — № 2. — С. 76-79. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 02.Dk, 02.30.Em. https://nasplib.isofts.kiev.ua/handle/123456789/82287 The method for defining basic geometric parameters of precision quadrupole lenses is offered. The method allows one to obtain the field of working area differing from required one less than by 10⁻⁴ compared with the basic component. The lens shape is described by 2 parameters. The method allows taking into account the field distortions caused by saturation of iron. Moreover, it is possible to calculate lenses with a complicated multipolar structure including the forbidden field harmonics. The paper contains the results of calculations for lense shapes used in actual installations. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Тheory and technics of particle acceleration Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation Расчет геометрических параметров работающей вблизи насыщения магнитной квадрупольной линзы Article published earlier |
| spellingShingle | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation Mytsykov, A.O. Тheory and technics of particle acceleration |
| title | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation |
| title_alt | Расчет геометрических параметров работающей вблизи насыщения магнитной квадрупольной линзы |
| title_full | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation |
| title_fullStr | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation |
| title_full_unstemmed | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation |
| title_short | Calculation of geometric parameters of a magnetic quadrupole lens working near to saturation |
| title_sort | calculation of geometric parameters of a magnetic quadrupole lens working near to saturation |
| topic | Тheory and technics of particle acceleration |
| topic_facet | Тheory and technics of particle acceleration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82287 |
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